Double Bruhat cells and total positivity

Abstract

We study the totally nonnegative variety<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript greater-than-or-equal-to 0"><mml:semantics><mml:msub><mml:mi>G</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">G_{\ge 0}</mml:annotation></mml:semantics></mml:math></inline-formula>in a semisimple algebraic group<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>. These varieties were introduced by G. Lusztig, and include as a special case the variety of unimodular matrices of a given order whose all minors are nonnegative. The geometric framework for our study is provided by intersecting<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript greater-than-or-equal-to 0"><mml:semantics><mml:msub><mml:mi>G</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">G_{\ge 0}</mml:annotation></mml:semantics></mml:math></inline-formula>with double Bruhat cells (intersections of cells of the two Bruhat decompositions of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>with respect to opposite Borel subgroups).